Note, that a poisson distribution does not automatically imply an exponential pdf for waiting times between events. This only accounts for situations in which you know that a poisson process is at work. But you'd need to prove the existence of the poisson distribution AND the existence of an exponential pdf to show that a poisson process is a suitable model!
As alluded to, this formula for P is found by composing the "nice" known solution for the Poisson kernel in the unit disk $\mathbb {D}$ through appropriate biholomorphisms so that it can be written as a function from $\tilde {A}$.
If two events occur at a rate of 1.8 per hour on average, and this occurrence follows a poisson process, what is the probability that there is at least 1 hour between two events? My approach for t...
Finding the probability of time between two events for a poisson process
The criteria in your book are for interval data; that would be useful if you had the dates at which hurricanes stroke... moreover these criteria are for constant rate Poisson processes, which is obviously (or I hope so) not the case of hurricanes. To check if your count data follow a Poisson distribution, a first elementary approach is the chi-square test.
How to know if a data follows a Poisson Distribution in R?
For Poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt {\lambda / n}$.
The distribution of $\ln u$ is exponential (see Wikipedia), and the Poisson distribution with parameter $\lambda$ counts the events per unit time in a Poisson point process with parameter $\lambda$, whose inter-point distances are indepedent exponential variables.